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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.extreme_dist"></a><a class="link" href="extreme_dist.html" title="Extreme Value Distribution">Extreme Value
        Distribution</a>
</h4></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">extreme</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
          <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">extreme_value_distribution</span><span class="special">;</span>

<span class="keyword">typedef</span> <span class="identifier">extreme_value_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">extreme_value</span><span class="special">;</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">extreme_value_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
   <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>

   <span class="identifier">extreme_value_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>

   <span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
   <span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>
</pre>
<p>
          There are various <a href="http://mathworld.wolfram.com/ExtremeValueDistribution.html" target="_top">extreme
          value distributions</a> : this implementation represents the maximum
          case, and is variously known as a Fisher-Tippett distribution, a log-Weibull
          distribution or a Gumbel distribution.
        </p>
<p>
          Extreme value theory is important for assessing risk for highly unusual
          events, such as 100-year floods.
        </p>
<p>
          More information can be found on the <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda366g.htm" target="_top">NIST</a>,
          <a href="http://en.wikipedia.org/wiki/Extreme_value_distribution" target="_top">Wikipedia</a>,
          <a href="http://mathworld.wolfram.com/ExtremeValueDistribution.html" target="_top">Mathworld</a>,
          and <a href="http://en.wikipedia.org/wiki/Extreme_value_theory" target="_top">Extreme
          value theory</a> websites.
        </p>
<p>
          The relationship of the types of extreme value distributions, of which
          this is but one, is discussed by <a href="https://www.google.com/books/edition/Extreme_Value_Distributions/GwBqDQAAQBAJ?hl=en&amp;gbpv=0" target="_top">Extreme
          Value Distributions, Theory and Applications Samuel Kotz &amp; Saralees
          Nadarajah</a>.
        </p>
<p>
          The distribution has a PDF given by:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">f(x) = (1/scale) e<sup>-(x-location)/scale</sup> e<sup>-e<sup>-(x-location)/scale</sup></sup></span>
          </p></blockquote></div>
<p>
          which in the standard case (scale = 1, location = 0) reduces to:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="serif_italic">f(x) = e<sup>-x</sup>e<sup>-e<sup>-x</sup></sup></span>
          </p></blockquote></div>
<p>
          The following graph illustrates how the PDF varies with the location parameter:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../graphs/extreme_value_pdf1.svg" align="middle"></span>

          </p></blockquote></div>
<p>
          And this graph illustrates how the PDF varies with the shape parameter:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../graphs/extreme_value_pdf2.svg" align="middle"></span>

          </p></blockquote></div>
<h5>
<a name="math_toolkit.dist_ref.dists.extreme_dist.h0"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.extreme_dist.member_functions"></a></span><a class="link" href="extreme_dist.html#math_toolkit.dist_ref.dists.extreme_dist.member_functions">Member
          Functions</a>
        </h5>
<pre class="programlisting"><span class="identifier">extreme_value_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">location</span> <span class="special">=</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">RealType</span> <span class="identifier">scale</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
</pre>
<p>
          Constructs an Extreme Value distribution with the specified location and
          scale parameters.
        </p>
<p>
          Requires <code class="computeroutput"><span class="identifier">scale</span> <span class="special">&gt;</span>
          <span class="number">0</span></code>, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">location</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          Returns the location parameter of the distribution.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">scale</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          Returns the scale parameter of the distribution.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.extreme_dist.h1"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.extreme_dist.non_member_accessors"></a></span><a class="link" href="extreme_dist.html#math_toolkit.dist_ref.dists.extreme_dist.non_member_accessors">Non-member
          Accessors</a>
        </h5>
<p>
          All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
          functions</a> that are generic to all distributions are supported:
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
        </p>
<p>
          The domain of the random parameter is [-∞, +∞].
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.extreme_dist.h2"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.extreme_dist.accuracy"></a></span><a class="link" href="extreme_dist.html#math_toolkit.dist_ref.dists.extreme_dist.accuracy">Accuracy</a>
        </h5>
<p>
          The extreme value distribution is implemented in terms of the standard
          library <code class="computeroutput"><span class="identifier">exp</span></code> and <code class="computeroutput"><span class="identifier">log</span></code> functions and as such should have
          very low error rates.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.extreme_dist.h3"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.extreme_dist.implementation"></a></span><a class="link" href="extreme_dist.html#math_toolkit.dist_ref.dists.extreme_dist.implementation">Implementation</a>
        </h5>
<p>
          In the following table: <span class="emphasis"><em>a</em></span> is the location parameter,
          <span class="emphasis"><em>b</em></span> is the scale parameter, <span class="emphasis"><em>x</em></span> is
          the random variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q
          = 1-p</em></span>.
        </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                  <p>
                    Function
                  </p>
                </th>
<th>
                  <p>
                    Implementation Notes
                  </p>
                </th>
</tr></thead>
<tbody>
<tr>
<td>
                  <p>
                    pdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: pdf = exp((a-x)/b) * exp(-exp((a-x)/b)) /
                    b
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: p = exp(-exp((a-x)/b))
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: q = -expm1(-exp((a-x)/b))
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: a - log(-log(p)) * b
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile from the complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: a - log(-log1p(-q)) * b
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mean
                  </p>
                </td>
<td>
                  <p>
                    a + <a href="http://en.wikipedia.org/wiki/Euler-Mascheroni_constant" target="_top">Euler-Mascheroni-constant</a>
                    * b
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    standard deviation
                  </p>
                </td>
<td>
                  <p>
                    pi * b / sqrt(6)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mode
                  </p>
                </td>
<td>
                  <p>
                    The same as the location parameter <span class="emphasis"><em>a</em></span>.
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    skewness
                  </p>
                </td>
<td>
                  <p>
                    12 * sqrt(6) * zeta(3) / pi<sup>3</sup>
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis
                  </p>
                </td>
<td>
                  <p>
                    27 / 5
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis excess
                  </p>
                </td>
<td>
                  <p>
                    kurtosis - 3 or 12 / 5
                  </p>
                </td>
</tr>
</tbody>
</table></div>
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